Collective Behavior of Swimming Bimetallic Motors

in Chemical Concentration Gradients.

by

Philip Matthew Wheat

A Dissertation Presented in Partial Fulfillment of the Requirements for the Degree

Doctor of Philosophy

Approved March 2011 by the Graduate Supervisory Committee:

Jonathan D. Posner, Chair

Patrick Phelan Kangping Chen Daniel Buttry

Ronald Calhoun

ARIZONA STATE UNIVERSITY

May 2011

i

ABSTRACT

Locomotion of microorganisms is commonly observed in nature. Although

microorganism locomotion is commonly attributed to mechanical deformation of

solid appendages, in 1956 Nobel Laureate Peter Mitchell proposed that an

asymmetric ion flux on a bacterium’s surface could generate electric fields that

drive locomotion via self-electrophoresis. Recent advances in nanofabrication

have enabled the engineering of synthetic analogues, bimetallic colloidal

particles, that swim due to asymmetric ion flux originally proposed by Mitchell.

Bimetallic colloidal particles swim through aqueous solutions by converting

chemical fuel to fluid motion through asymmetric electrochemical reactions.

This dissertation presents novel bimetallic motor fabrication strategies,

motor functionality, and a study of the motor collective behavior in chemical

concentration gradients. Brownian dynamics simulations and experiments show

that the motors exhibit chemokinesis, a motile response to chemical gradients that

results in net migration and concentration of particles. Chemokinesis is typically

observed in living organisms and distinct from chemotaxis in that there is no

particle directional sensing. The synthetic motor chemokinesis observed in this

work is due to variation in the motor’s velocity and effective diffusivity as a

function of the fuel and salt concentration. Static concentration fields are

generated in microfluidic devices fabricated with porous walls. The development

of nanoscale particles that swim autonomously and collectively in chemical

concentration gradients can be leveraged for a wide range of applications such as

directed drug delivery, self-healing materials, and environmental remediation.

ii

ACKNOWLEDGEMENTS

I would like to thank my committee comprised of Dr. Patrick E. Phelan,

Dr. Kangping Chen, Dr. Daniel A. Buttry, and Dr. Ronald J. Calhoun for all of

their help and guidance from my comprehensive exams through my dissertation

defense. I especially thank my advisor and committee chair Dr. Jonathan D.

Posner for inviting me to be a part of his research team and for his guidance and

support throughout my graduate studies at Arizona State University.

I also thank my colleagues in the Posner Research Group who have been

great friends and have always been selfless in providing ideas, instruction, and

assistance in my work. Specifically I thank Abishek Jain for teaching me how to

use the tools in the clean room and how to make PDMS structures. I also want to

specifically acknowledge Steve Klein, Guru Navaneetham, Kamil Salloum, Juan

Tibaquira, Carlos Perez, Wen-Che Hou, and Charlie Corredor for thoughtful

discussions and input throughout my studies. I especially thank Jeffrey Moran and

Nathan Marine for their tremendous collaborative effort in my publications

accompanying this research. I also thank Babak Yagoupi for staying up late

Friday nights making structures with me for the experiments.

I would like to thank the staff in CSSER and CSSS for their instruction

and for the use of their facilities throughout my research, especially Grant

Baumgardner and Karl Weiss for their help with sample preparation and electron

microscopy, and Carrie Sinclair for all of her help securing clean room supplies.

I am extremely greatful to the Marines at the Phoenix Officer Selection

Office for their continual efforts to accommodate my studies, specifically Captain

iii

Mark Beasely, Gunnery Sergeant Edgar Arriaga, and Master Sergeant Mike

Edmonds (Ret.).

I am eternally grateful to my wife Stacy, and my daughters Kaitlyn,

Danielle, and Amanda, who bore the greatest burden of my long hours in the lab

and at home studying for the last five and a half years. I also thank my mother,

Charlene, for all of her help with the kids.

Finally I wish to thank my father, Dr. Stephen R. Wheat. Without his

guidance, encouragement, and support it is doubtful I would have embarked on

this journey.

iv

TABLE OF CONTENTS

Page

LIST OF FIGURES ............................................................................................... vi

CHAPTER

1. INTRODUCTION ...................................................................................... 1

1.1 Motivation .................................................................................... 1

1.2 Literature Review ..................................................................................... 1

2.

1.3 Significance .................................................................................. 4

BACKGROUND ........................................................................................ 6

2.1 Nanomotors .................................................................................. 6

2.2 Synthetic Nanomotors .................................................................. 8

2.3 Directional Control ..................................................................... 10

2.4 Chemical Gradients for Directional Control .............................. 12

2.5 Chemotaxis-Chemokinesis Terminology ................................... 13

2.6 Biological Chemotaxis ............................................................... 21

2.7 Biological Chemokinesis ............................................................ 23

2.8 Synthetic Chemotaxis and Chemokinesis................................... 23

2.9 Chemotactic Assays ................................................................... 26

2.10 ChemotacticMeasures ............................................................... 34

2.11 Bimetallic Nanomotors ............................................................. 36

v

CHAPTER Page

3.

2.12 BimetallicNanomotorEfficiency ............................................... 37

THEORETICAL FRAMEWORK ............................................................ 40

3.1 Analytical Approach .................................................................. 40

4.

3.2 ComputationalApproach ............................................................. 51

EXPERIMENTAL METHODS ................................................................ 54

4.1 Fabrication of Rod-Shaped Nanomotors .................................... 54

4.2 Fabrication of Spherical Motors ................................................. 59

4.3 Synthetic Chemokinesis Assays ................................................. 68

5.

4.4 Experimental Apparatus ............................................................. 74

RESULTS AND DISCUSSION ............................................................... 88

5.1 Brownian Dynamics Simulation Results: ................................... 88

5.2 Variable Diffusion PDE Model: ................................................. 98

6.

5.3 Experimental Results ................................................................ 103

SUMMARY ............................................................................................ 128

APPENDIX

REFERENCES…………………………………………………………….....129

A MATLAB SIMULATION OPTIONS…..…………………………..133

B BROWNIAN DYNAMICS CODE.................................................... 148

C COPYRIGHT RELEASE AGREEMENTS………………………...163

vi

LIST OF FIGURES

Figure Page

1. Motion of motor proteins. ................................................................................... 6

2. F1-ATPase modified nanopropellar. ................................................................... 7

3. Bacteria driven micro-rotator….......................................................................... .8

4. Tethered Au-Ni nanorotor…………………………………………………… 9

5. Bimetallic nanomotor hauling cargo…………………….…………………. 10

6. Magnetic field directed Au-Ni-Au-Pt nanorods ............................................... 11

7. Leukocytes oriented along a gradient ............................................................... 16

8. Depiction of different types of chemotaxis…. .................................................. 17

9. One-dimensional depiction of an orthokinetic cell ........................................... 19

10. One-dimensional depiction of an orthokinetic cell without walls .................. 20

11. Conceptual design for chemotaxis nanomotor ................................................ 24

12. Depiction of a nanomotor containing nickel. .................................................. 25

13. A depiction an Au-Ni-Au-Pt nanomotor attached to cargo ............................ 26

14. Boyden chamber. ............................................................................................ 30

15. Flow cell design .............................................................................................. 32

16. Schematic of a gold/platinum nanomotor. ...................................................... 37

17. Effective diffusivity as a function of position. ................................................ 48

18. Chemotactic index as a function of time. ........................................................ 49

19. Bimetallic nanorod fabrication process. ......................................................... 55

20. Exploded view of the electrochemical cell ..................................................... 57

21. Schematic of the fabrication method. ............................................................. 62

vii

Figure Page

22. Scanning electron microscope image .............................................................. 63

23. Representative traces for 3 µm microspheres ................................................. 67

24. Average bimetallic, spherical micromotor speeds. ......................................... 68

25. Nanomotor speed as a function of the electrical resistance. ........................... 70

26. Electrochemical modulation of bimetallic nanomotor speed. ......................... 71

27. Experimental apparatus used by Calvo-Marzal et al. ..................................... 71

28. Interdigitated working and counter electrode ................................................. 72

29. Concentration profiles of salt .......................................................................... 75

30. Schematic of the structure ............................................................................... 76

31. Exploded view of the gradient generator ........................................................ 79

32. Initial channel structure design. ...................................................................... 82

33. Generation 2 channel design. .......................................................................... 82

34. Generation 3 channel design. .......................................................................... 83

35. Generation 4 channel design. .......................................................................... 83

36. Generation 5 channel design. .......................................................................... 84

37. Final channel design. ...................................................................................... 84

38. Micromotor speeds versus H2O2 .................................................................... 89

39. Average polystyrene sphere speed versus H2O2 concentration ...................... 89

40. Average speed versus silver salt concentration. ............................................. 90

41. The inverse of the average rotational diffusivity ............................................ 90

42. Initial distribution of nanomotors ................................................................... 92

43. Final steady-state distribution of nanomotors ................................................. 94

viii

Figure Page

44. Chemotactic index phase diagram. ................................................................. 94

45. Chemotactic index versus time. ...................................................................... 95

46. Response time vs chemotactic velocity at maximum fuel concentration. ...... 96

47. Normalized motor concentration .................................................................. 100

48. Normalized motor concentration 2 ............................................................... 101

49. Comparison between PDE and Brownian dynamics .................................... 102

50. Experimental set-up for microscopy ............................................................. 104

51. Average nanomotor velocity as a function of position ................................. 106

52. Example of a discretized nanomotor path. .................................................... 110

53. Total displacement squared ........................................................................... 110

54. Mean squared displacement .......................................................................... 111

55. Average nanomotor effective diffusivity. ..................................................... 111

56. Channel regions used to measure chemotactic index. .................................. 112

57. Chemotactic index measured as a function of time ...................................... 113

58. Vertically-averaged fluorescence intensity ................................................... 115

59. Vertically-averaged fluorescence intensity integrated horizontally ............. 115

60. Initial distribution of nanomotors subject to a linear gradient in KCl. ......... 117

61. Distribution of nanomotors subject to a linear gradient. ............................... 117

62. Histogram contour map ................................................................................. 118

63. Case 1 results. ............................................................................................... 119

64. Case 2 results.. .............................................................................................. 120

65. Case 3 results. ............................................................................................... 120

ix

Figure Page

66. Case 4 results ................................................................................................ 121

67. Case 5 results. ............................................................................................... 121

68. Case 6 results. ............................................................................................... 122

69. Case 7 results. ............................................................................................... 122

70. Experimental, Brownian dynamics simulation, PDE model. ....................... 123

71. Steady state chemotactic index phase map. .................................................. 125

72. Response time phase map. ............................................................................ 126

73. Steady state chemotactic index vs. grad(Deff) x w /Deff,min. ..................... 126

1

CHAPTER 1

INTRODUCTION

1.1 Motivation

Synthetic nanomotors are of particular interest in the research community because

of their potential ability to mimic biological nanomotors. In many cases,

biological nanomotors are responsible for delivering cargo to very specific

destinations in biological systems. Synthetic nanomotors have been developed

that are capable of picking up, transporting, and dropping off cargo. Unfortunately

a sufficient method of steering the nanomotors to a specific location has not been

developed. Often biological cells utilize variations in the chemical concentrations

in their immediate vicinity to move to very specific locations. If synthetic

nanomotors were developed capable of responding to chemical concentration

gradients as a means of passively guiding them to a destination, it would be a

tremendous step in realizing the use of nanomotors for applications such as highly

specific drug delivery. There are three distinct locomotive responses to chemical

concentration gradients: chemotaxis, chemokinesis, and diffusiophoresis. While

chemotaxis and chemokinesis are commonly leveraged in biological systems, to

date there is no account of a demonstration of either synthetic chemotaxis or

synthetic chemokinesis in the literature.

1.2 Literature Review of Synthetic Nanomotor Responses to Chemical

Gradients

Chemotaxis, since its discovery as a means of guiding the direction of motion by

2

Engelmann in 1881,(Engelmann 1881) has been the topic of more than 22,000

publications. The primary means of determining chemotactic behavior of a cell

has been the observation of the global response of large numbers of the cells.

Unfortunately, it is quite possible to mistake a global accumulation of cells at the

source of a chemical as a chemotactic response, when in actuality it is a purely

random diffusive type response. This mistake has been made so frequently in the

literature that several articles have been written in attempts to address the

pervasive underlying misconceptions that lead to this mistake. In 1973, Zigmond

et al. established a crude method of distinguishing between the purely random

response and a chemotactic response that is only applicable for a specific type of

assay.

In 2007, Hong et al. claimed to demonstrate “the first experimental example of

chemotaxis outside biological systems” using synthetic bimetallic

nanomotors.(Hong et al. 2007) They used two different types of assay to

demonstrate this. First they used the capillary assay in which a capillary is filled

with hydrogen-peroxide, capped at one end and then placed in an aqueous

solution containing several bimetallic nanomotors. In this case, the evidence of a

chemotactic response is the mild accumulation of nanomotors in the capillary

over time. The second assay used a gel plug that was saturated with hydrogen-

peroxide and then placed in an aqueous solution containing the synthetic

nanomotors. In this case, the evidence of a chemotactic response is the global

motion of the nanomotors predominantly towards the gel plug. Finally, the

authors back up their claim using Brownian dynamics simulations. In the case of

3

the first assay, it is impossible to say that the accumulation of a small number of

the nanomotors in a capillary containing hydrogen-peroxide is the result of

chemotactic behavior as oppose to random motion. The authors argue that the

increased speed due to the higher concentration of hydrogen-peroxide is necessary

for the nanomotors, which are otherwise scurrying along the lower surface of the

chamber, to climb up over the lip of the capillary. However, the increased speed

would have the same effect on a nanomotor that just happens to be in the region

as a result of random motion.

However, if the global behavior of the nanomotors is governed by a purely

random process, then the results of the second assay are counter-intuitive. As the

nanomotors approach the hydrogen-peroxide saturated gel plug, they should move

faster and quickly move away, resulting in higher dwell times at lower

concentrations such that the equilibrium distribution of nanomotors shows an

accumulation at regions of lower hydrogen-peroxide concentration. Instead, what

is reported is an accumulation at the gel plug. There are at least two possible

explanations for this discrepancy. First, the nanomotors become stuck in the

vicinity of the gel plug and remain there through the duration of the experiment,

such that over time the nanomotors accumulate in the vicinity of the gel plug.

Second, there is actual chemotaxis taking place in which the nanomotors have a

directional bias towards the region of higher concentration. It is difficult to

imagine where such a bias might originate when dealing with a simple bimetallic

nanorod. If it were due to surface irregularities resulting from the non-precise

fabrication process, then one would expect to observe a substantial number of the

4

nanomotors displaying an opposite bias, down the gradient. Fortunately, there is a

video accompanying these results. Upon inspection, one can see that the

nanomotors drift towards the gel plug regardless of whether they are oriented

towards or away from the gel plug. This is a clear indication that the experiment is

invalidated by the presence of a pressure driven or otherwise generated

superimposed flow, or the global behavior observed may be a dominating

diffusiophoretic response. Furthermore, the authors’ supporting Brownian

Dynamics simulation admittedly incorporated a slight bias directed toward higher

hydrogen-peroxide concentrations. Such a bias is necessary for a chemotactic

response, but again is not characteristic of bimetallic nanomotors.

1.3 Significance

Here, I present Brownian dynamic simulations I use to argue that the global

behavior of synthetic nanorods, as currently, constructed is limited to

chemokinesis and without modification will not exhibit any form of chemotactic

response. Furthermore, I experimentally validate these conclusions. For this work,

I fabricated two different types of bimetallic nanomotors. First I use the

traditional bimetallic nanorods that I fabricated using the methods prescribed in

the literature.(Paxton et al. 2004) Then I use bimetallic spherical motors that I

fabricated using a technique that I developed and recently published in

Langmuir.(Wheat et al.) My experimental approach utilizes the structure

conceived by Diao et al. and used by Palacci et al. for the purpose of studying

diffusiophoresis.(Palacci et al. 2010) This approach provides substantial

5

improvement over most chemotaxis and chemokinesis assays in that it generates a

static spatial chemical concentration gradient without flow. Using these

experiments I demonstrate the first case of synthetic chemokinesis. The Brownian

Dynamics model can be used to predict the chemokinetic component of a

perceived chemotactic response in both biological and synthetic systems. Finally,

results of the model are reduced to a partial differential equation that can be

solved rapidly for a quantitative analysis of the global behavior of chemokinetic

cells.

6

CHAPTER 2

BACKGROUND

2.1 Nanomotors

The term nanomotor refers to an object less than a micrometer in one or more

spatial dimension that takes a form of non-mechanical energy and converts it into

mechanical work. Biological nanomotors, sometimes referred to as molecular

motors, have long been known to exist in the form of protein motors and nucleic

acid motors. Nucleic acid motors include RNA polymerases, which transcribe

RNA from DNA, and DNA polymerases, which produce double-stranded DNA

from single stranded DNA. Protein motors include myosins, which are

responsible for muscle contractions, kinesins, which carry cargo along

microfilaments within a cell, and dyneins, which are responsible for ciliary and

flagellar motility.(Bloom 1996) Figure 1 depicts the typical motion of the

different types of motor proteins.

Figure 1: Motion of motor proteins and an F1-ATPase rotator. Hess, H., Bachand, G. D. & Vogel, V.

Powering nanodevices with biomolecular motors. Chemistry-a European Journal 10, 2110-2116 (2004).

Copyright Wiley-VCH Verlag GmbH & Co. KGaA. Reproduced with permission.

7

The existence and functionality of dynein were accurately predicted as early as

1965(Gibbons 1965) and were proven in 1987.(Paschal, Shpetner and Vallee

1987) In the following decade work was done to characterize protein motors,

such as determining the force a kinesin protein is capable of exerting.(Meyhofer

and Howard 1995) During the late 1990’s and early 2000’s, research efforts

shifted towards incorporating protein motors into synthetic systems to create

functionally specific, hybrid bio-synthetic nanomotors.(van den Heuvel and

Dekker 2007) In one instance, a synthetic nanorod was attached to an F1

Figure 2

-ATPase

(a protein dubbed factor F1 that synthesizes Adenosine Triphosphate) motor to

create a hybrid bio-synthetic nanopropellar ( ).(Soong et al. 2000) In

another instance, a microrotor powered by bacteria was created by confining

unidirectionally swimming bacteria in a rotor track (Figure 3).(Hiratsuka et al.

2006)

Figure 2: F1-ATPase modified nanopropellar. From [Soong, R. K. et al. Powering an inorganic

nanodevice with a biomolecular motor. Science 290, 1555-1558 (2000)]. Reprinted with permission from

AAAS. http://dx.doi.org/10.1126/science.290.5496.1555

http://dx.doi.org/10.1126/science.290.5496.1555�

8

Figure 3: Bacteria driven micro-rotator. Hiratsuka, Y., Miyata, M., Tada, T. & Uyeda, T. Q. P. A

microrotary motor powered by bacteria. Proceedings of the National Academy of Sciences of the United

States of America 103, 13618-13623 Copyright (2006) National Academy of Sciences, U.S.A.

http://dx.doi.org/10.1073/pnas.0604122103

2.2 Synthetic Nanomotors

Within the past decade, efforts have shifted towards the development of fully

synthetic nanomotors in an effort to take advantage of the relative experimental

simplicity associated with non-biological environments. In 2004, Paxton et al.

discovered the autocatalytic motion of bimetallic nanorods in the presence of

hydrogen peroxide.(Paxton et al. 2004) In this seminal work, the nanorods were

370 nm in diameter with adjoined 1 μm gold and 1 μm platinum segments. In 2 to

3% hydrogen-peroxide the nanomotor velocities were on the order of 10 body

lengths per second. Paxton et al. observed the dimensions and velocities to be

comparable to multiflageller bacteria.

http://dx.doi.org/10.1073/pnas.0604122103�

9

Shortly thereafter, Fournier-Bidoz et al. published their discovery of a different

bimetallic combination that also exhibited autonomous motion in the presence of

hydrogen-peroxide.(Fournier-Bidoz et al. 2005) In the Fournier-Bidoz et al.

paper, the nanorods were half gold and half nickel, with one side tethered to a

substrate these nanorods behaved as nanorotors, pivoting around the attachment

point. These papers sparked significant research interest in bimetallic nanomotors.

In 2005, Catchmark et al. produced gold gears 150 µm in diameter with a

platinum coating on only one side of each cog, resulting in autonomous rotation in

the presence of hydrogen-peroxide.(Catchmark, Subramanian and Sen 2005) In

2008, we collaborated with Burdick, Laocharoensuk, and Wang to demonstrate

nanomotors capable of picking up, hauling, and releasing micron-scale

cargo.(Burdick et al. 2008) Sundararajan et al. demonstrated similar capabilities

the same year. (Sundararajan et al. 2008)

Figure 4: Tethered Au-Ni nanorotor. [Fournier-Bidoz, S., Arsenault, A. C., Manners, I. & Ozin, G. A.

Synthetic self-propelled nanorotors. Chemical Communications, 441-443 (2005).]– Reproduced by

permission of The Royal Society of Chemistry http://dx.doi.org/10.1039/b414896g

http://dx.doi.org/10.1039/b414896g�

10

Figure 5: Bimetallic nanomotor hauling cargo. Adapted with permission from Sundararajan, S.,

Lammert, P. E., Zudans, A. W., Crespi, V. H. & Sen, A. Catalytic motors for transport of colloidal cargo.

Nano Letters 8, 1271-1276. Copyright 2008 American Chemical Society.

http://dx.doi.org/10.1021/nl072275j

2.3 Directional Control

Currently significant research efforts in the nanomotor field are focused on

directional control, particularly the ability to guide the nanomotors to a specific

destination for the purpose of delivering cargo or for self-assembly processes.

Their motion can be controlled using external magnetic fields(Sundararajan et al.

2008; Burdick et al. 2008) as well as chemical(Calvo-Marzal et al. 2009; Ibele,

Mallouk and Sen 2009; Hong et al. 2007) and thermal(Balasubramanian et al.

2009) fields. In 2008, we demonstrated the use of magnetic fields to guide gold-

nickel-gold-platinum nanorods through a PDMS channel network.(Burdick et al.

2008)

http://dx.doi.org/10.1021/nl072275j�

11

Figure 6: Magnetic field directed Au-Ni-Au-Pt nanorods through a PDMS channel network. Adapted

with permission from Burdick, J., Laocharoensuk, R., Wheat, P. M., Posner, J. D. & Wang, J. Synthetic

nanomotors in microchannel networks: Directional microchip motion and controlled manipulation of

cargo. Journal of the American Chemical Society 130, 8164-+ Copyright 2008 American Chemical

Society. http://dx.doi.org/10.1021/ja803529u

Calvo-Marzal et al. demonstrated the ability to accelerate and decelerate

nanomotors by varying local oxygen concentrations in the presence of electric

fields sufficiently small to preclude electrophoretic effects, but large enough to

electrochemically affect the local concentrations of oxygen.(Calvo-Marzal et al.

2009) Autonomous micromotors composed of AgCl have been shown to

asymmetrically decompose when exposed to UV illumination resulting in local

chemical concentration gradients inducing a diffusiophoretic response. In this

case the angle and intensity of the illumination can be manipulated to control the

global behavior of the motors. Balasubramanian et al. demonstrated a linear

relationship between temperature and bimetallic nanorod speed and the ability to

thermally modulate the speed of nanomotors.(Balasubramanian et al. 2009)

http://dx.doi.org/10.1021/ja803529u�

12

2.4 Chemical Gradients for Directional Control

For most perceived future nanomotor applications, it would be ideal to guide the

nanomotors to their destination without the use of externally applied fields. Two

distinct types of such passive guidance that have been observed in biological

systems in response to chemical concentration gradients are chemotaxis and

chemokinesis. For the purposes of drug delivery, it would be ideal for nanomotors

carrying cargo to passively seek out the location in the body where the drugs are

needed. It has been shown that surface wounds emanate hydrogen-peroxide, and it

is suspected that the resulting gradient in hydrogen-peroxide concentration is the

signal that guides leukocytes to the wound for healing. If nanomotors could be

engineered to swim up such gradients, it is conceivable that drugs aiding in the

healing of a wound could navigate to the wound in response to the increase in

hydrogen-peroxide concentration at that location. Thus far, very limited work has

been done to determine the chemotactic potential of synthetic nanomotors.(Hong

et al. 2007) Furthermore, while there has been an enormous amount of research

on biological chemotaxis and chemokinesis over the past 122 years, there is still a

lot that is not well understood. Basic questions, such as whether or not active-

directional sensing is a necessary component for chemotaxis, persist in the

literature.(Hong et al. 2007) There is a lot of confusion in the field on terminology

and how to distinguish between the purely random responses to chemical

gradients characteristic of chemokinesis and the directional sensing nature of

chemotaxis.

13

2.5 Chemotaxis-Chemokinesis Terminology

In 1881, Engelmann first postulated the concept of chemotaxis, wherein a cell

would navigate to sources of chemical concentration gradients.(Engelmann 1881)

This type of navigation was first observed in bacteria by Pfeffer and in

Leukocytes by Leber in 1888.(Pfeffer 1888; Leber 1888) Since then, there have

been more than 22,000 papers on the topic of chemotaxis. Many of the papers

focus on what cells exhibit chemotaxis and what chemicals trigger chemotactic

responses. Other papers focus on the fundamental mechanisms of chemotaxis in

efforts to determine how cells sense gradients, whether or not temporal or spatial

sensing are required, whether or not cells or the chemicals they are sensing or

both have to be surface bound, etc. However, in 1973, Zigmond et al. pointed out

a fundamental flaw in many of the preceding assays used to determine whether or

not cells exhibited a chemotactic response to certain chemicals.(Zigmond and

Hirsch 1973) In nearly all of the chemotactic assays, the chemotactic response

was measured by analyzing the end state distribution of cells. For example, if

cells were initially distributed uniformly across two regions, one region having a

particular chemical in question, and the other region not having the chemical, and

then the cells tended towards a non-uniform equilibrium, then the cells were said

to have responded chemotacticly to the chemical. However, as Zigmond points

out, a non-uniform equilibrium response could be due to purely random motion.

For example, if a cell swims faster in higher concentrations of a certain chemical

but its orientation is random, then the motion will continue to be random in the

presence of a concentration gradient of that chemical. If that cell and the chemical

14

concentration gradient are both located in a bounded region with reflective walls,

then the region within the chamber with the lowest chemical concentration will

ultimately be the region with the greatest accumulation of cells. This

accumulation is because the cells go slower in the region of low concentration and

end up having a higher dwell time in that region than in the high concentration

regions where they speed through. This accumulation is a response to the

chemical concentration gradient, but it is a response that comes about because of

purely random motion and can not to be considered chemotaxis. In 1981, Dunn

wrote a chapter in a book edited by Wilkinson underscoring the importance of

making this distinction between purely random motion, which he calls

chemokinesis, and chemotaxis.(Dunn 1981) As Dunn points out, if the walls of

the bounded region were removed, these particular cells would diffuse infinitely

to a concentration of nearly zero everywhere including the region with the low

chemical concentration. On the other hand, if a cell were exhibiting true

chemotactic behavior, when the bounding walls are removed, the cell will

eventually find its way towards or away from the source of the chemical

concentration gradient, depending on whether it is attracted or repelled by that

chemical. In chemotaxis, containing walls are not required for the accumulation

of cells at a source or sink of a particular chemical concentration this allows for

the process to be a long distance process where a single cell will eventually reach

its destination. In chemokinesis, the strength of the accumulation of cells is

dependent on the proximity of the walls, and at steady state there is a non-uniform

pseudo-equilibrium distribution throughout the bounded region with cells

15

randomly moving in and out of the region of accumulation.

Both Dunn’s 1981 chapter and Wilkinson’s 1998 review article on chemotaxis

attempt to realign the field of chemotaxis with the following internationally

accepted, but far too frequently neglected, terminology:

Chemotaxis is where chemical substances, more specifically gradients in

concentration of chemical substances, alone determine the direction of

locomotion. A form of directional sensing is absolutely necessary for chemotaxis.

This can be accomplished using spatial or temporal sensing mechanisms. A

chemotactic response cannot come about by purely random locomotion. An

additional point of confusion in the literature is the mistaken idea that the

orientation of the cell or object must be in alignment with gradients in chemical

concentration, as is the case with leukocytes, see Figure 7. E-coli, on the other

hand, will travel with a constant speed in some arbitrary direction, then stop,

reorient (or tumble), and then travel (or run) at that same speed in a new direction

(termed “run and tumble” behavior). The frequency of tumbling events increases

as the bacteria moves down the gradient in concentration of certain chemicals.

The bacteria use a temporal sensing mechanism to determine the current

concentration is greater than a previous concentration. As a result, once the E-coli

bacteria reach a peak in concentration, they begin tumbling very frequently

because every direction results in a decrease in concentration. The net result being

that they migrate towards a peak in concentration where they linger. In this case,

the orientation of the cell is random with persistence (which comes about from the

temporal sensing mechanism) in the direction of increasing concentration.

16

Figure 7: Leukocytes oriented along a gradient in chemoattractant concentration, with the source to the right of the image. Reprinted from Zigmond, S. H. "Ability of Polymorphonuclear Leukocytes to Orient in Gradients of Chemotactic Factors." Journal of Cell Biology

75.2: 606-16., Copyright (1977), with permission from Elsevier.

17

Figure 8: Depiction of different types of chemotaxis, a) gradient aligned migration (as is the case with

leukocytes), b) random walk behavior with a temporal sensing mechanism such that the rate of turning

increases when the object is moving down the gradient and decreases when moving up the gradient (as

is the case with E. coli), c) biased random motion with a persistence in the direction of increasing

concentration.

18

Chemokinesis is where chemical substances determine the speed and/or turning

frequency (or rotational diffusivity). The subcategories of chemokinesis are

orthokinesis where the speed is only determined by the chemical substances and

klinokinesis where only the turning frequency is determined by the chemical

substances. Furthermore, the changes in speed or turning frequency corresponding

to changes in chemical concentration are termed chemokinetic responses. It is

possible, and in many cases necessary, for an object undergoing chemotaxis to

exhibit chemokinetic responses.

To further clarify the distinction between chemotaxis and chemokinesis, consider

a one dimensional scenario with a single object subject to a chemical gradient in a

region bound by two reflective walls as shown in Figure 9. First consider a cell

that exhibits orthokinesis. The cell travels in one direction until it encounters a

boundary and then it travels in the other direction until it encounters the other

boundary, and so on. The chemical present causes the object to travel slow in high

concentrations and fast in low concentrations. There is a linear concentration

gradient with very low concentrations on the left side and very high

concentrations on the right side. As a result, the object moving from the right wall

to the left wall travels very slowly at first and then speeds up and quickly moves

through the region to the left side encountering the wall and quickly moving back

towards the right when it begins to slow down again and very slowly approaches

the right wall. Over time, the object spends an equal time moving up the gradient

as it does moving down the gradient. However, the object clearly spends more

time located in the region of high concentration on the right where it is moving

19

very slowly. Furthermore, if one were to extend this example to include several

objects that do not interact with each other, then an accumulation of the objects

would appear in the region of higher concentration. This sort of behavior is

exactly what is often mistaken for chemotaxis. In reality, this is chemokinesis.

Figure 9: One-dimensional depiction of an orthokinetic cell bound by reflective walls on the left and

right subject to a chemical concentration gradient, where the velocity decreases with an increase in

concentration.

Now consider a second scenario in which the same region is used, but the object

in question utilizes a directional sensing mechanism that causes it to move very

fast when the chemical concentration is increasing and very slowly when the

chemical concentration is decreasing. Also modify the object such that it turns

around at regular time intervals. As a result, the object will move large distances

to the right when facing the right, and then when facing the left it will not move

very far at all, resulting in a ratcheting motion to the right. Even though half of the

time the object is oriented to the left, its net motion is always to the right, where it

will linger when it reaches the maximum concentration. Again, if this example

were extend to include several objects that do not interact with each other, then an

accumulation of the objects would appear in the region of higher concentration. In

this case, the observed behavior is chemotaxis.

20

In both scenarios, the objects spend more time on the right side of the chamber.

As a result, both objects may appear to exhibit chemotaxis. An obvious distinction

is that chemotaxis results in a continued migration towards the region of

accumulation, whereas chemokinesis ultimately reaches a non-uniform pseudo

equilibrium distribution. Unfortunately this distinction is less obvious

experimentally where most objects exhibiting chemotaxis have a substantial

portion of the population that is defective and does not respond to the chemical

concentration gradient. The defective population causes the end-state distribution

to appear as a non-uniform pseudo-equilibrium distribution. However, if the

objects truly exhibited chemotaxis, then if the chemical concentration were

mirrored about the right wall and both walls were removed, as shown in Figure

10; the objects would find their way to the region of maximum chemical

concentration. However, if the walls, which are the only means of reorienting the

object in the first scenario, are removed in the first scenario, then the object will

clearly wander increasingly far from the maximum chemical concentration. In the

second scenario, the object would still work its way towards the maximum in

chemical concentration, exhibiting true chemotaxis.

Figure 10: One-dimensional depiction of an orthokinetic cell without walls subject to a chemical

concentration gradient, where the velocity decreases with an increase in concentration.

21

2.6 Biological Chemotaxis

Several different types of cells have been observed to respond to the presence of

certain chemicals, called chemoattractants (or chemokines if they are proteins

secreted from cells), by working their way up gradients in concentration of that

chemical, seeming to seek out or forage for regions of maximum concentration.

This behavior is referred to as positive chemotaxis. In some cases the cells

migrate down gradients in chemical concentration, the chemical in this case is

frequently referred to in the literature as a toxin or a chemorepellent. In such

cases, the behavior is referred to as negative chemotaxis.

The most studied cell that exhibits chemotaxis is Escherichia coli (E-coli) which

propels itself using flagellar motors and works its way up concentration gradients

of chemicals such as MeAsp (α-methyl-DL-aspartate) and down concentration

gradients of chemicals such as NiCl2.(Mello and Tu 2007; Sourjik and Berg

2002) The motion of chemotactic bacteria is typically characterized as random

walk. The E-coli bacteria swim in relatively straight lines for periods in which the

flagellar motors are rotating in one direction, and then the bacteria tumble and

rotate relatively quickly when the motors are reversed. The tumbling motion has

the effect of reorienting the bacteria such that subsequent straight motion will be

in a different direction. The time between tumbling events appears random with a

dependence on the gradient in local concentrations of attractants and repellents.

When the concentration of a chemoattractant increases or a chemorepellent

decreases, the bacteria swim straight for longer periods (i.e. have lower turning

frequencies). As the concentration of a chemoattractant decreases or a

22

chemorepellent increases, the turning frequency increases reducing the movement

in less favorable directions.

The E-coli bacteria has been shown to have a temporal sensing mechanism that

initiates changes in turning frequency based on receptor binding events.(Brown

and Berg 1974) At each moment in time, the cell compares the current

chemoattractant concentration with the concentration from some previous time. If

the current concentration is lower than the previous, the turning frequency

increases and vice versa. The velocity of the E-coli bacteria during the run portion

of the run and tumble behavior is independent of the chemoattractant

concentration. Different chemotaxiing cells have been observed with

fundamentally different chemokinetic responses. The chemotaxiing planktonic

bacteria P. haloplanktis increases velocity and turning frequency with increasing

chemoattractant concentrations.(Seymour et al. 2008) In both cases, positive

chemotaxis is observed, with entirely different responses to increased

concentrations. Leukocytes have an entirely different chemotactic mechanism as

well. Leukocytes, which are otherwise spherically shaped, elongate when exposed

to chemoattractants. They swim in the direction that their long axis points in, and

that direction is random in the absence of a gradient in the chemoattractant

concentration. When subject to a gradient in the chemoattractant concentration the

long axes align with the gradient and the cells swim up the gradient. These cells

also have a distinctly orthokinetic response, accompanying their directional

sensing abilities. As the local chemoattractant concentration increases, so does the

translational velocity.

23

2.7 Biological Chemokinesis

For every three papers focused on biological chemotaxis there has been one

focused on biological chemokinesis. Chemokinesis has sparked less interest

because it is not an effective means of long distance navigation. However, in

many cases chemotaxis has been shown not to be the cause of observed biological

migration. In 2007, Inamdar et al. identified the primary purpose of the jelly coat

of a sea urchin egg is to locally increase the motility of the sperm and thereby the

sperm-egg collision frequency. The response of sperm to the extracellular jelly

coat is purely chemokinetic without any directional sensing component. In this

case a chemical concentration gradient is established to guide cells via

chemokinesis. Other cells have been shown to exhibit chemokinesis including

human sperm,(Ralt et al. 1994) human neural cells,(Richards et al. 2004) and

several types of bacteria.(Brown et al. 1993)

2.8 Synthetic Chemotaxis and Chemokinesis

It has been suggested that synthetic nanomotors exhibit chemotaxis in fuel

concentration gradients.(Hong et al. 2007) At low concentrations of hydrogen-

peroxide (less than 5 wt %), both spherical and rod-shaped nanomotor exhibit a

chemokinetic response as their velocities have been shown to increase roughly

linearly with an increase in hydrogen –peroxide.(Laocharoensuk, Burdick and

Wang 2008; Wheat et al.) This chemokinetic relationship parallels the biological

response of certain cells that exhibit chemotaxis and all cells that exhibit

24

chemokinesis, but does not involve any directional sensing. Therefore, the

bimetallic nanomotors can be expected to mimic biological motors that utilize

chemokinesis. However, in order for synthetic, bimetallic, nanomotors to exhibit

chemotaxis, they would have to incorporate some form of temporal or spatial

concentration gradient sensing abilities. This is clearly not present in the case of

the simple bimetallic nanorods. However, it is possible to design synthetic motors

in a way that effectively incorporates a spatial sensing capacity. Consider a

bimetallic rod modified as shown in Figure 11 with a non-conducting segment

leading to a smaller perpendicularly oriented bimetallic segment. This

perpendicular segment would induce a rotational component that will cause the

motor to rotate faster when the tail is in higher concentrations, and slower in

lower concentrations such that as the nanomotor circles around it moves further

when facing up the gradient than it does when facing down the gradient. The end

result would be a ratcheting behavior up the concentration gradient.

Figure 11: Conceptual design for a gold-platinum nanomotor that would undergo chemotaxis in a

hydrogen-peroxide concentration gradient, the black segment is non-conducting.

25

In previous work, we joined nanomotors containing a nickel segment to

polystyrene spheres with super paramagnetic iron oxide nanocrystal shells, as

shown in Figure 12.(Burdick et al. 2008) In this work, I present a method of

coating polystyrene spheres such that half of the surface is covered with one

metal, and the other half with another, creating a bimetalic spherical motor. One

approach to realizing a synthetic motor capable of directional sensing as depicted

in Figure 11, would be to coat a magnetic sphere to make it a bimetallic motor,

and then join that to a nanorod containing a nickel segment, as shown in Figure

13. Such a combination would result in a ratcheting behavior up a chemical

concentration gradient and could be the first case of synthetic chemotaxis.

Figure 12: Depiction of a nanomotor containing a nickel segment joined to a polystyrene sphere with a super paramagnetic iron oxide nanocrystal shell.

26

Figure 13: A depiction an Au-Ni-Au-Pt nanomotor attached to a magnetic microsphere that is half coated in gold and half coated in platinum, resulting in a chemotaxis capable synthetic motor.

2.9 Chemotactic Assays

There are a variety of assays for studying chemotaxis and chemokinesis, but until

recently, none of these were ideal. An ideal assay incorporates a steady gradient

in chemoattractant concentration and the ability for the object being tested for a

chemotactic response to pass from low concentration to high concentration and

then back down again without trapping the object.

One type of chemotactic assay utilizes a capillary containing the chemoattractant

and capped at one end. The capillary is placed in a solution containing the

27

chemotactic object (cell or motor), and the chemoattractant diffuses out of the

capillary into the solution setting up a transient gradient in fuel

concentration.(Hong et al. 2007) If the diffusion of the attractant is slow relative

to the response of the cells or motors, then the capillary provides a local, high

attractant concentration. When the attractant diffuses to motors or cells capable of

positive chemotaxis, the motors or cells will migrate up the concentration gradient

and into the capillary. Over time, all chemotacticly functional motors or cells will

accumulate in the capillary. At even longer times, the attractant will diffuse

towards a uniform distribution throughout the system, and the motors or cells will

also diffuse back to a uniform distribution. If the motors or cells exhibit negative

chemotaxis (i.e. the attractant is a toxin/repellant), then all of the functional

motors or cells will migrate towards the regions of the system far away from the

capillary opening where the concentration is lowest.

If the motors or cells exhibit positive orthokinesis and negative (or no)

klinokinesis they will, and the system is enclosed, then the motors will migrate

with asymmetric diffusion towards a non-uniform equilibrium distribution, with

an accumulation at the low attractant concentration region far away from the

capillary opening. It is important to note this is distinct from the negative

chemotaxis case because in this case there will still be motors migrating both up

and down the concentration gradient and at equilibrium there will be no net flux

of motors or cells and there will be motors or cells in the capillary.

28

The appeal of this assay is its simplicity. It is very easy to fill a capillary, cap an

end, and place it in a solution containing motors or cells. Furthermore, the lip of

the capillary reduces the number of motors that enter by pure random motion, as

the thickness of the side walls of the capillary must be overcome by random

vertical displacement. For cells or motors that generally settle and move along the

lower surface of a chamber, very few will enter the capillary without a

deterministic motion up the gradient. This barrier makes it easier to distinguish

between chemotaxis and negative orthokinetic and positive (or no) klinokinetic

response.

This approach is less than ideal and cannot be used to adequately analyze the

chemotactic ability of bimetallic-nanomotors in hydrogen-peroxide for two

reasons. First, the diffusivity of the hydrogen-peroxide is much higher than the

effective diffusivity of the bimetallic-nanomotors resulting in a transient gradient.

Second, the capped capillary does not allow for the nanomotors to pass through

the high concentration and move back into lower concentrations without turning

around. The turnaround time results in artificially high dwell times at the higher

concentrations, an effect that is difficult to distinguish from a chemotactic

response. Using Brownian Dynamic simulations we showed that trapping a

motor in a high or low attractant concentration region greatly increases the

concentration of motors in that region even if the motors or cells do not exhibit

chemotaxis or chemokinesis in response to the attractant, and the motion of the

motors or cells is governed purely by diffusion. Furthermore, a cell or motor that

29

exhibits positive orthokinetic response to an attractant will accumulate away from

the region of high concentration, but if the motor or cell is impeded or temporarily

trapped in the high concentration region, the accumulation may occur in the high

concentration region.

In 1962, Boyden developed an assay specifically designed to study the

chemotactic response of cells that behave like leukocytes.(Boyden 1962) Such

cells are initially spherical and become elongated in the direction of motion when

subjected to a chemoattractant. The assay consists of a chamber (now called the

Boyden chamber depicted in Figure 14) divided into two regions by a filter

designed such that the pores are too narrow for the spherical shaped cells to pass

through, but large enough for the cells in the elongated configuration to pass

through. One region contains the chemoattractant and the other region contains

the cells. The two regions behave as reservoirs such that the chemoattractant

concentration is assumed to develop a linear gradient through the depth of the

filter. Since then, variations of the Boyden chamber have been the primary

method for studying biological chemotaxis. This method is attractive because it is

relatively easy to set up multiwell plates where each well is an individual Boyden

chamber for high throughput screening of chemicals that may incite either

chemokinesis or chemotaxis for a particular motor or cell. Unfortunately there is

no way to observe the motion of the motors or cells within the gradients. As has

been pointed out on a number of occasions by Zigmond, Dunn, and Wilkinson,

observing an end state accumulation of cells in the region containing the

30

chemoattractant does not allow for a distinction between chemotaxis and

chemokinesis.(Zigmond and Hirsch 1973; Wilkinson 1998; Dunn 1981) This

approach can only be done to analyze chemotaxis if the chemokinetic responses

are fully characterized and used to predict the response that is due to

chemokinesis. A deviation from this response would imply chemotaxis.

Figure 14: Boyden chamber.

In general, in a bound system, it is difficult to distinguish between a non-uniform

pseudo-steady state accumulation of motors or cells due to chemokinesis and a

non-uniform distribution that arises due to chemotaxis with a chemotactic

potential < 100%. The chemotactic potential is the percent of chemotactic motors

or cells in a sample of that are functional. The most straight forward approach to

distinguish between chemotaxis and chemokinesis is to visualize the motion of

the motors or cells in the gradient. If the motors systematically work their way up

or down the gradient, then an observed accumulation is likely chemotaxis. On the

other hand, if the motors traverse high and low concentration region multiple

times in the development of the accumulation, then the accumulation is the result

of chemokinesis. One approach that allows for migration visualization is the use

of flow cells. Flow cells are a widely used alternative approach that allows for a

31

steady chemoattractant gradient.(Saadi et al. 2006; Lin and Butcher 2006) The

flows cells are microfluidic devices that funnel two inlets to a single channel

where the cells are allowed to diffuse as shown in Figure 15.(Lin and Butcher

2006) This design results in a spatially steady concentration gradient that can be

leveraged if the flow of the chemoattractant has a much higher Peclet number

than the flow of the chemotactic objects or cells. Such a scenario is achieved if

either the diffusivity of the chemotactic object is much higher than the diffusivity

of the chemoattractant, or the downstream velocity of the chemotactic object is

much lower than the downstream velocity of the chemoattractant. The latter is

achieved by using cells that are adsorbed to the surface of the flow cell and have a

minimal Stokes-drag profile.(Lin and Butcher 2006) This approach cannot be

applied to the bimetallic nanomotors produced to date because they become

completely fixed when adsorbed to channel surfaces and otherwise have a much

slower diffusivity than hydrogen-peroxide and have non-negligible Stokes drag

such that they advect downstream with the velocity of the flow. Either the

chemotactic object needs to be faster than diffusion if it is freely swimming or if

the cell is adhered to the bottom plate then it is relatively unaffected by the flow.

Nanomotors are freely swimming so in that case you need their effective

diffusivity to be higher than the diffusivity of the chemoattractant. Unfortunately

the effective diffusivity of the motors at the highest speeds achieved to date is an

order of magnitude less than the diffusivity of hydrogen peroxide.

32

Figure 15: Flow cell design used by Lin and Butcher to measure the chemotactic response of T cells to

various chemokines. Lin, F., and E. C. Butcher. "T Cell Chemotaxis in a Simple Microfluidic Device."

Lab on a Chip 6.11 (2006): 1462-69. – Reproduced by permission of The Royal Society of Chemistry

http://dx.doi.org/10.1039/b607071j

In 2008, Seymor et al. used a three stream flow cell in which the center channel

introduced a relatively slow diffusing chemoattractant, and the outer two channels

introduced a salt water solution containing fast swimming oceanic planktic

bacteria. In this case, the migration of the bacteria up the chemoattractant

concentration gradient is much faster than the diffusion of the chemoattractant.

This is necessary for the survival of the bacteria that forage for nutrients in

diffusing patches often caused by cells lysing. If the motors or cells steer or align

http://dx.doi.org/10.1039/b607071j�

33

along chemoattractant gradients the typical nanomotor velocity might be

sufficient to observe chemotaxis using this assay. In 10 seconds, H2O2

will

diffuse approximately 140 μm. In this case, a motor would have to travel faster

than 14 μm/s up the gradient to observe appreciable chemotaxis.

Ahmed and Stocker developed a chemotactic assay based on a valved channel

containing a high concentration chemoattractant reservoir at one end and an

opening to a perpendicular flow channel containing extremely high concentrations

of the E-coli bacteria.(Ahmed and Stocker 2008) With the valve closed, the side

channel experiences no advective flow, only diffusion of the attractant into and

the bacteria out of the perpendicular flow channel. While this approach can have a

more steady fuel concentration gradient than the capillary assay, it still suffers

from the higher dwell time effect of the single capped end. This assay is effective

for chemotaxis because the accumulation can be distinguished from a

chemokinetic accumulation. However, if one is interested in study the

chemokinesis of an object, the single capped end and the sink/source flow end do

not allow for an observed accumulation to be attributed to chemokinesis because

the higher dwell times at the capped end will result in an accumulation

independent of a chemokinesis.

In 2008, Palacci et al. successfully generated steady concentration gradients in a

microfluidic channel structure for the purpose of studying

diffusiophoresis.(Palacci et al. 2010) Palacci et al. generated the steady gradient

34

using a method first introduced by Diao et al. in 2006.(Diao et al. 2006) Diao et

al.’s design incorporates three parallel channels in a porous membrane. The

membrane allows for solution diffusion but resists pressure driven flow. By

flowing an aqueous solution containing a solute species in one of the outer two

channels and an aqueous solution without the solute species in the other outer

channel while the solution in the center channel remains stationary, the outer two

channels act as a source-sink pair. This configuration results in a steady linear

gradient of solute concentration in the center channel. This approach is ideal for

both chemotaxis and chemokinesis assays as it allows visualization of the objects

throughout the assay, and there are no restraints on the response time of the

objects relative to the diffusivity of the chemoattractants.

2.10 Chemotactic Measures

The primary measure of chemokinsesis or chemotaxis used in the literature is the

chemotactic index (CI). The chemotactic index is typically given as the ratio of

the number chemotactic objects or cells in a region containing the maximum

concentration of the chemoattractant to the number of objects or cells in a region

of equal size that contains minimum (typically zero) chemoattractant

concentration. The problem with this definition is that if there is a complete

depletion of motors in the low concentration region, then the CI approaches

infinite. Also if there is a very small number of motors in the low concentration

region then the CI becomes very sensitive to motors entering and leaving the low

35

accumulation region, resulting a very noisy measure of chemotactic index.

Alternatively, the CI has been defined as the ratio of chemoactive objects in the

high concentration region divided by the normalized average number of objects.

Others have observed individual cell behavior and have used more advanced

calculations to determine a chemotactic sensitivity χ, a parameter that measures a

populations attraction to a specific chemical intrinsically.(Ahmed and Stocker

2008) In this case, a model developed by Rivero et al. for the flux of

chemotaxiing bacteria results in the following equation:

𝜒 =tanh−1�3𝜋𝑉𝑐8𝜈 �𝜋8𝜈

𝐾𝐷�𝐾𝐷+𝐶�

2𝑑𝐶𝑑𝑥

,

where KD is the dissociation constant for the bacteria receptors and the specific

chemoattractant, which is previously know from reaction kinetics

experiments.(Rivero et al. 1989) Vc

is the net speed of the population up the

gradient, ν is the translational speed of the individual cells, and C is the local

chemoattractant concentration. Each of these values, and the gradient in chemical

concentration, are measured for several different concentrations in order to

determine χ. This equation is only valid for a specific type of chemotactic

behavior, particularly the klinokinesis exhibited by E-coli. The advantage is that

the chemotactic sensitivity is a measure of chemotactic response to an attractant

that is independent of the actual local attractant concentration gradient.

36

2.11 Bimetallic Nanomotors

A bimetallic nanomotor in an aqueous solution containing hydrogen peroxide

results in hydrogen peroxide oxidation at the anodic end generating oxygen

molecules, protons, and electrons. The electrons generated conduct through the

nanomotor and combine with protons, hydrogen peroxide, and oxygen to

complete the reduction reaction at the cathodic end. This process is depicted in

Figure 16 for a nanorod composed of gold (cathode) and platinum (anode). The

reactions result in a local excess in protons at the anodic end and a local depletion

of protons at the cathodic end. The gradient in proton concentration within the

surrounding electrolyte leads to an asymmetric charge density and ultimately an

electric field directed from the anodic end to the cathodic end, as shown in Figure

16. The electric field coupled with the charge density produces an electrical body

force driving the surrounding fluid from the anode to the cathode. In a reference

frame where the fluid is stationary, this fluid motion translates to the locomotion

of the nanomotor with the anode forward. The fundamental mechanism of motion

resembles electrophoresis; however in this case, the electric field and the charge

density distribution are generated by particle. The underlying physics of

bimetallic motors has been studied extensively by Moran et al.(Moran and Posner

2011; Moran, Wheat and Posner "Locomotion of Electrocatalytic Nanomotors

Due to Reaction Induced Charge Autoelectrophoresis" 2010)

37

Figure 16: Schematic of a gold/platinum nanomotor of typical dimensions depicting the

electrochemical reactions that occur at each end, as well as the resulting charge density and the

resulting electric field lines. The red region denotes high charge density due to the local excess of

protons generated at the anodic surface and the blue region denotes the low charge density due to the

depletion of protons at the cathodic surface.(Moran, Wheat and Posner "Locomotion of

Electrocatalytic Nanomotors Due to Reaction Induced Charge Auto-Electrophoresis" 2010)

http://pre.aps.org/abstract/PRE/v81/i6/e065302

2.12 Bimetallic Nanomotor Efficiency

The efficiency of bimetallic motors can be calculated from the theoretical Stokes

drag, the measured velocity, current density measurements, and the average Gibbs

free energy of the electrochemical reactions. The efficiency η is the ratio of the

mechanical power output to the chemical power input. The mechanical power

output can be calculated as a product of the force exerted and the speed attained.

Because the speed remains relatively constant, the nanomotor is assumed to be in

equilibrium with the force exerted in equilibrium with the Stokes drag on the

motor. The magnitude of Stokes drag for a cylinder can be approximated

analytically by treating the cylinder as an ellipsoid, in this case the Stokes drag is

given by

http://pre.aps.org/abstract/PRE/v81/i6/e065302�

38

𝐹 = 4𝜋𝜇𝑐𝑢ln�2𝑐𝑏 �−

12,

where µ is the viscosity, c is half the length of the cylinder, b is the radius, and u

is the speed.(K. A. Rose et al. 2007) For b = 0.11 µm, c = 1 µm, µ = 1x10-3 N

s/m2, and u = 25 µm/s (for 6wt% H2O2) ,(Wheat et al. 2010) F = 0.17 pN.

Therefore the mechanical power output is uF = 4.25x10-18

The chemical power input can be calculated as product of the reaction flux j, the

surface area A of the motor, and the total Gibbs free energy ∆G of the reactions.

The reaction flux is calculated from the published current density for

electrochemical decomposition of 6wt% H

W.

2O2 at a gold platinum interface, which

is i = 0.684 C/s m2

𝑗 = 𝑖𝑛𝐹

,

.(Paxton, Kline et al. 2006) The reaction flux is given by

where n = 2 is the number of electrons transferred in the reaction and F = 96 485

C/mol is the Farraday constant. The surface area of a 220 nm radius, 2mm long

nanomotor is 1.3823 µm2

Figure 16

. The total Gibbs free energy for the reaction is

determined using tables.(Moore 2010)The primary reactions are depicted in

. Oxidation of the H2O2 occurs at the platinum end resulting in the

products 2H+, O2, and 2e-. Reduction occurs at the gold end with H2O2 + 2H+ +

2e- resulting in 4H2O. The only species involved in the reaction with non-zero

standard energies of formation (∆G0f) are H2O (∆G0f,H20 = -237.2 KJ/mol) and

H2O2 (∆G0f,H202 = -114.0 KJ/mol).(Moore 2010) Assuming reduction and

39

oxidation occur at the same rate, the total Gibbs free energy of the reactions is ∆G

= -720.8 KJ/mol, or the energy available from the reactions is 720.8 KJ/mol.

Therefore the chemical power input is jA∆G = 3.5x10-12

η = 𝑃𝑚𝑒𝑐ℎ𝑃𝑐ℎ𝑒𝑚

= 4.25x10−18 W

3.5x10−12 W= 1.2𝑥10−6.

W. Finally, the efficiency

is

40

CHAPTER 3

THEORETICAL FRAMEWORK

3.1 Analytical Approach

In order to predict the ability of synthetic nanomotors to exhibit global behavior

analogous to biological chemokinesis, we model chemokinesis as a purely

diffusive response. To do this, it is necessary to represent a random walk behavior

with an effective diffusivity. Such a representation is used by Howse et al. to

model the behavior of a self-motile Janus sphere, using the following equation:

𝐷𝑒𝑓𝑓 = 𝐷 +14𝑈𝑎𝑑𝑣2

𝐷𝑟𝑜𝑡,

where Deff is the effective diffusivity, Uadv is the translational velocity, Drot is the

rotational diffusivity, and D is the diffusivity due to Brownian motion, or the

diffusivity in the absence of a chemical promoting a chemokinetic

response(Howse et al. 2007). Chemokinetic responses imply that the advective

velocity and/or rotational diffusivity are functions of a chemical concentration,

i.e. Uadv = f1(Cfuel/nutrients) for orthokinesis and/or Drot = f2(Cfuel/nutrients) for

klinokinesis. From the above equation for effective diffusivity, it is expected that

a chemokinetic response can be expressed in terms of the effective diffusivity as a

function of chemical concentration, i.e. Deff = f(Cfuel/nutrients). From there, spatial

variations in concentration will translate directly to spatial variations in effective

diffusivity. As a result, the flux of chemokinetic objects in a spatial gradient of

fuel/nutrient concentration can be expressed using the generalization of Fick’s law

that deals with significant spatial variations in diffusivity for a Brownian walker.

41

Consider a one-dimensional Brownian walker or particle that has a direction

speed component u and a turning frequency f. The rotation of the particle is

random such that half way through turning around, an individual particle is

equally likely to complete the direction change as it is to return to the original

direction. In this case the average frequency of a direction change is f/2. Let R be

the number density of particles moving right along the one-dimensional (x) axis,

and L be the number density of particles moving left. Conservation of the particles

can be written as:

𝜕𝑅𝜕𝑡

= −𝜕𝑢𝑅𝜕𝑥

+ 𝑓2𝐿 − 𝑓

2𝑅,

and

𝜕𝐿𝜕𝑡

= 𝜕𝑢𝐿𝜕𝑥

− 𝑓2𝐿 + 𝑓

2𝑅.

The total number density of particles (ρ) is R+L, and the particle flux (J) is u(R–

L). Adding the conservation equations yields:

𝜕𝑅𝜕𝑡

+ 𝜕𝐿𝜕𝑡

= −𝜕𝑢𝑅𝜕𝑥

+ 𝜕𝑢𝐿𝜕𝑥

⇒ 𝜕(𝑅+𝐿)𝜕𝑡

= −𝜕𝑢(𝑅−𝐿)𝜕𝑥

⇒ 𝜕𝜌𝜕𝑡

= − 𝜕𝐽𝜕𝑥

,

and subtracting the two equations yields

𝜕𝑅𝜕𝑡− 𝜕𝐿

𝜕𝑡= −𝜕𝑢𝑅

𝜕𝑥− 𝜕𝑢𝐿

𝜕𝑥+ 𝑓𝐿 − 𝑓𝑅 ⇒ 𝜕(𝑅−𝐿)

𝜕𝑡= −𝜕𝑢(𝑅+𝐿)

𝜕𝑥− 𝑓(𝑅 − 𝐿),

⇒ 𝜕𝜕𝑡�𝐽𝑢� = −𝜕𝑢𝜌

𝜕𝑥− 𝑓 𝐽

𝑢.

Multiplying by velocity and differentiating w.r.t. x yields:

𝜕𝜕𝑥�𝑢 𝜕

𝜕𝑡�𝐽𝑢�� = − 𝜕

𝜕𝑥�𝑢 𝜕𝑢𝜌

𝜕𝑥� − 𝜕

𝜕𝑥(𝑓𝐽).

Let u be a function of x and not a function of time, such that

42

𝜕𝜕𝑥� 𝜕𝜕𝑡

(𝐽)� = − 𝜕𝜕𝑥�𝑢 𝜕𝑢𝜌

𝜕𝑥� − 𝜕

𝜕𝑥(𝑓𝐽).

For the L.H.S, the order of differentiation is interchangeable.

𝜕𝜕𝑡�𝜕𝐽𝜕𝑥� = − 𝜕

𝜕𝑥�𝑢 𝜕𝑢𝜌

𝜕𝑥� − 𝜕

𝜕𝑥(𝑓𝐽).

Recall,

𝜕𝐽𝜕𝑥

= −𝜕𝜌𝜕𝑡

, such that the conservation equation becomes:

𝜕2𝜌𝜕𝑡2

= + 𝜕𝜕𝑥�𝑢 𝜕𝑢𝜌

𝜕𝑥� + 𝜕

𝜕𝑥(𝑓𝐽).

For diffusive processes for which short time behavior is of little interest, the

second derivative in time can be considered negligible. The resulting equation,

𝜕𝜕𝑥�𝑢 𝜕𝑢𝜌

𝜕𝑥�+ 𝜕

𝜕𝑥(𝑓𝐽) = 0 ⇒ 𝜕(𝑓𝐽) = −𝜕 �𝑢 𝜕𝑢𝜌

𝜕𝑥�,

is integrated to yield:

∫𝜕(𝑓𝐽) = −∫𝜕 �𝑢 𝜕𝑢𝜌𝜕𝑥 � ⇒𝑓𝐽 = −𝑢𝜕𝑢𝜌𝜕𝑥

,

⇒𝐽 = −𝑢𝑓𝜕𝑢𝜌𝜕𝑥

.

There are 5 variations of this equation worth discussing.

Case 1: u is a constant and f is a constant. In this case the flux equation becomes:

𝐽 = −𝑢2

𝑓𝜕𝜌𝜕𝑥

.

Let u2/f be defined as the effective diffusivity Deff

𝐽 = −𝐷𝑒𝑓𝑓𝜕𝜌𝜕𝑥

,

of the particles. Here the flux

equation becomes the traditional Fick’s law of diffusion:

where Deff

is a constant.

43

Case 2: u is a constant and f varies in space. In this case the flux equation

becomes:

𝐽 = − 𝑢2

𝑓(𝑥)𝜕𝜌𝜕𝑥

= −𝐷𝑒𝑓𝑓(𝑥)𝜕𝜌𝜕𝑥

.

In this case, the effective diffusivity varies with space, such that the flux equation

is Fick’s law of diffusion for variable diffusivity. It is clear that the steady state

equilibrium particle distribution is a uniform distribution.

Case 3: The ratio u(x)/f(x) is a constant. In this case, the flux equation can be

expanded using the product rule:

𝐽 = −𝑢𝑢(𝑥)𝑓

𝜕𝜌𝜕𝑥− 𝜌 𝜕

𝜕𝑥�𝑢𝑢(𝑥)

𝑓� = −𝐷𝑒𝑓𝑓

𝜕𝜌𝜕𝑥− 𝜌 𝜕𝐷𝑒𝑓𝑓

𝜕𝑥= 𝜕

𝜕𝑥�𝜌𝐷𝑒𝑓𝑓�.

Here the flux equation is the Fokker-Plank law of diffusive flux.

Case 4: u varies with x, while f is a constant. In this case, the flux equation can be

written as:

𝐽 = −𝑢2(𝑥)𝑓

𝜕𝜌𝜕𝑥− 𝑢(𝑥)𝜌 𝜕

𝜕𝑥�𝑢(𝑥)

𝑓�.

From the product rule:

𝑢(𝑥)𝜌 𝜕𝜕𝑥�𝑢(𝑥)

𝑓� + 𝑢(𝑥)𝜌 𝜕

𝜕𝑥�𝑢(𝑥)

𝑓� = 𝜌 𝜕

𝜕𝑥�𝑢

2(𝑥)𝑓�

⇒𝑢(𝑥)𝜌 𝜕𝜕𝑥�𝑢(𝑥)

𝑓� = 𝜌

2𝜕𝜕𝑥�𝐷𝑒𝑓𝑓�.

⇒ 𝐽 = −𝐷𝑒𝑓𝑓𝜕𝜌𝜕𝑥− 𝜌

2𝜕𝜕𝑥�𝐷𝑒𝑓𝑓�.

The previous cases are described in detail by Schnitzer. The following cases are

my work.

44

Case 5: u and f vary independently with x. In the most general case, both speed

and turning frequency will vary with position. In this case, the flux equation is:

𝐽 = −𝑢2(𝑥)𝑓(𝑥)

𝜕𝜌𝜕𝑥− 𝑢(𝑥)

𝑓(𝑥)𝜌 𝜕𝑢(𝑥)

𝜕𝑥.

Again from the product rule:

𝜌 𝜕𝜕𝑥�𝑢

2

𝑓� = 𝜌𝑢2 𝜕

𝜕𝑥�1𝑓� + 𝜌

𝑓𝜕𝜕𝑥

(𝑢2) =𝜌𝑢2 𝜕𝜕𝑥�1𝑓�+ 2𝜌𝑢

𝑓𝜕𝑢𝜕𝑥

,

such that

𝜌𝑢𝑓𝜕𝑢𝜕𝑥

= 𝜌2𝜕𝜕𝑥�𝑢

2

𝑓� − 𝜌𝑢

2

2𝜕𝜕𝑥�1𝑓�.

As a result, the flux equation becomes:

𝐽 = −𝐷𝑒𝑓𝑓𝜕𝜌𝜕𝑥− 𝜌

2𝜕𝐷𝑒𝑓𝑓𝜕𝑥

+ 𝜌𝑢2

2𝜕𝜕𝑥�1𝑓�.

In this case, the flux equation does not reduce to a form of the Fokker-Plank law,

and information about turning frequency is necessary to determine the flux. Note

that the diffusive flux equations in cases 2, 3 and 4 can be generalized as follows:

𝐽 = −𝐷𝑒𝑓𝑓𝜕𝜌𝜕𝑥− 𝛼𝜌 𝜕𝐷𝑒𝑓𝑓

𝜕𝑥.

This flux equation is referred to as the modified Fokker-Planck law of diffusive

flux, where α is the Ito-Stratonovich coefficeint. In cases 2, 3, and 4, α = 0, 1, and

0.5 respectively. For the more general case 5, the flux equation can only be

reduced to this form if

𝜌2𝜕𝐷𝑒𝑓𝑓𝜕𝑥

− 𝜌𝑢2

2𝜕𝜕𝑥�1𝑓� = 𝛼𝜌 𝜕𝐷𝑒𝑓𝑓

𝜕𝑥,

which requires

𝜕𝜕𝑥�1𝑓� = −�2𝛼−1

𝑢2� 𝜕𝐷𝑒𝑓𝑓

𝜕𝑥.

45

This requirement can be written in a more insightful way. Recall from the product

rule expansion above,

𝜌𝑢2

2𝜕𝜕𝑥�1𝑓� = 𝜌

2𝜕𝜕𝑥�𝑢

2

𝑓� − 𝜌𝑢

𝑓𝜕𝑢𝜕𝑥

= 𝜌2𝜕𝐷𝑒𝑓𝑓𝜕𝑥

− 𝜌𝑢𝑓𝜕𝑢𝜕𝑥

,

which reduces to:

𝜕𝜕𝑥�1𝑓� = 1

𝑢2𝜕𝐷𝑒𝑓𝑓𝜕𝑥

− 2𝑢𝑢2𝑓

𝜕𝑢𝜕𝑥

.

Therefore the following equation must be satisfied to reduce the flux equation to

the Ito-Stratonovich convention:

1𝑢2

𝜕𝐷𝑒𝑓𝑓𝜕𝑥

− 2𝑢𝑢2𝑓

𝜕𝑢𝜕𝑥

= −�2𝛼−1𝑢2

� 𝜕𝐷𝑒𝑓𝑓𝜕𝑥

,

⇒ 𝜕𝐷𝑒𝑓𝑓𝜕𝑥

− 2𝑢𝑓𝜕𝑢𝜕𝑥

= −(2𝛼 − 1) 𝜕𝐷𝑒𝑓𝑓𝜕𝑥

.

⇒ 𝑢𝑓𝜕𝑢𝜕𝑥

= 𝛼 𝜕𝐷𝑒𝑓𝑓𝜕𝑥

.

Case 5a: Any value of alpha can be obtained if the above equation is true.

Solving for alpha this equation becomes:

𝛼 = 𝑢𝑓𝜕𝑢𝜕𝑥

𝜕𝐷𝑒𝑓𝑓𝜕𝑥

.

Otherwise the flux equation remains in the following general form:

𝐽 = −𝐷𝑒𝑓𝑓𝜕𝜌𝜕𝑥− 𝜌

2𝜕𝐷𝑒𝑓𝑓𝜕𝑥

+ 𝜌𝑢2

2𝜕𝜕𝑥�1𝑓�,

which requires knowledge of the turning frequency and the axial velocity to

determine the flux.

46

Case 5b: Now consider a less general case with a constant α, and a constant 𝜕𝑢𝜕𝑥

,

𝑢𝑓∝ 𝜕𝐷𝑒𝑓𝑓

𝜕𝑥.

Recall that if the left hand side is a constant, then α = 1. If the left hand side is not

constant, an alternative value of α is obtained.

Case 5c: Alternatively, consider a linear gradient in Deff

𝑢𝑓𝜕𝑢𝜕𝑥

= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡.

. In this case,

For the PDE analysis in this work, the modified Fokker-Planck law of diffusive

flux is used:

𝐽 = −𝐷𝑒𝑓𝑓𝜕𝜌𝜕𝑥− 𝛼𝜌 𝜕𝐷𝑒𝑓𝑓

𝜕𝑥,

where the turning frequency f is analogous to a rotational diffusivity. In order to

use this equation, a value must be determined for α. This value was selected by

considering the underlying physics of the motors, and by comparison with the

Brownian Dynamics simulations, discussed in section 3.2. In previous work, I

have observed a linear relationship between nanomotor speed and fuel

concentration (Figure 24).(Wheat et al. 2010) Furthermore, our research team has

not yet had success in measuring a klinokinetic response for the nanomotors.

Therefore I program the Brownian Dynamics simulation to account for a linear

relationship between speed and fuel concentration, and no variation in rotational

diffusivity. Recall from the above discussion, that this situation corresponds to

case 4, where α = 0.5. To compare the PDE model to the BD simulation, the BD

47

simulation was first run with a linear spatial gradient in speed (simulating a linear

gradient in H2O2 and a linear relationship between the motor speed and the H2O2

Figure 17

concentration). The effective diffusivity of the simulated motors was calculated as

a function of position; the results are shown as red circles in . The input

into the PDE model is the blue fit curve shown in Figure 17. The simulation and

model results are compared in Figure 18. The red line shows the BD. The blue

line shows the PDE (alpha=0.5) with the gradient in diffusivity determined from

the BD as shown in blue in figure 24. It is clear from Figure 18, that the PDE

model under these conditions is not capturing all of the behavior in the BD

simulation. Specifically there is some variation in rotational diffusivity that is

imposed by the reflective wall boundary condition which results in a reduction in

the effective diffusivity ne